Event segment

Event Segments

An event is a label that abstracts a change. Given an event set $Z$, the null event denoted by $\epsilon \not \in Z$ stands for nothing change. The time base of our concerning systems is denoted by $\mathbb {T}$, and defined

$\mathbb {T} =[0,\infty )$

as non-negative real numbers. The time base plus infinity is denoted by $\mathbb {T} ^{\infty }$, and defined

$\mathbb {T} ^{\infty }=[0,\infty ]$

as non-negative real numbers plus infinity.

A timed event $(z,t)$ over an event set $Z$ and a time interval $[t_{l},t_{u}]\subseteq \mathbb {T}$ denotes that an event $z\in Z$ occurs at time $t\in [t_{l},t_{u}]$. The null segments over time interval $[t_{l},t_{u}]$ is denoted by $\epsilon _{[t_{l},t_{u}]}$ which means that there is no event over $[t_{l},t_{u}]$.

If there exists one event $z\in Z$ at time $t\in [t_{l},t_{u}]$, we call it an unit event segments. More precisely, the unit event segment $(z,t)$ over an event set $Z$ and time interval $[t_{l},t_{u}]$ is equivalent to

$\epsilon _{[t_{l},t]}(z,t)\epsilon _{[t,t_{u}]}$

where $\epsilon _{[t_{l},t]}$ and $\epsilon _{[t,t_{u}]}$ are respectively called pre-null segment and post-null segment of the unit event segment.

Given an event set $Z$, concatenation of two unite event segments $\omega$ over $[t_{1},t_{2}]$ and $\omega '$ over $[t_{3},t_{4}]$ is denoted by $\omega \omega '$ whose time interval is $[t_{1},t_{4}]$, and implies $t_{2}=t_{3}$. A multi-event segment $(z_{1},t_{1})(z_{2},t_{2})\cdots (z_{n},t_{n})$ is concatenations of unite event segments $\epsilon _{[t_{l},t_{1}]}(z_{1},t_{1}),\epsilon _{[t_{1},t_{2}]}(z_{2},t_{2})$ and $\epsilon _{[t_{n-1},t_{n}]}(z_{n},t_{n})\epsilon _{[t_{n},t_{u}]}$ where $t_{l}\leq t_{1}\leq t_{2}\leq \cdots \leq t_{n-1}\leq t_{n}\leq t_{u}$.

The {universal timed language over an event set $Z$ and a time interval $[t_{l},t_{u}]\subseteq \mathbb {T}$, is denoted by $\Omega _{Z,[t_{l},t_{u}]}$, and is defined as the set of all possible event segments. Formally,

$\Omega _{Z,[t_{l},t_{u}]}=\{(z,t)^{*}|z\in Z,t\in [t_{l},t_{u}]\}$

where $^{*}$ denotes a none or multiple concatenation(s) of timed events. Notice that the number of events in a string $\omega \in \Omega _{Z,[t_{l},t_{u}]}$ can be either of zero, finite or infinite. Infinite many events in a string $\omega \in \Omega _{Z,[t_{l},t_{u}]}$ implies that $t_{u}-t_{l}=\infty$, however $t_{u}-t_{l}=\infty$ does not imply infinite many events in it.

A language over an event set $Z$ and a timed interval $[t_{l},t_{u}]$ is a set of timed strings over $Z$ and $[t_{l},t_{u}]$. If $L$ is a language over $Z$ and $[t_{l},t_{u}]$, then $L\subseteq \Omega _{Z,[t_{l},t_{u}]}$.

References

• [Zeigler76] Bernard Zeigler (1976). Theory of Modeling and Simulation (first ed.). Wiley Interscience, New York.
• [ZKP00] Bernard Zeigler, Tag Gon Kim, Herbert Praehofer (2000). Theory of Modeling and Simulation (second ed.). Academic Press, New York. ISBN 978-0127784557.{{cite book}}: CS1 maint: multiple names: authors list (link)